Nonlinear State Equations Research Articles

A Novel Wiener‐Type Dynamic Neural Network Method for Large Signal Modeling of Power Amplifiers

ABSTRACTA novel Wiener‐type dynamic neural network (Wiener‐type DNN) method for large signal modeling of power amplifiers (PAs) is proposed in this paper. In this method, the proposed model structure for the PA contains two Wiener‐type DNNs to describe the input impedance and the amplification efficiency, respectively. The Wiener‐type DNN includes a simplified linear dynamic equation module and a nonlinear static equation module. For the simplification process of the linear dynamic equation, it is proposed to process fundamental frequency components of the large signal harmonic data of the PA by vector fitting. The neural network is used to implement the static equation in the Wiener‐type DNN. The formulas for training the proposed Wiener‐type DNN model of the PA using large signal data are derived. The training algorithm of the PA model is proposed to improve the training efficiency. A Motorola MOSFET PA example and a Freescale lateral double‐diffused MOSFET PA example are present to validate the proposed Wiener‐type DNN method. For the Motorola MOSFET PA, a dynamic neural network (DNN) model and a time‐delay deep neural network (TDDNN) are established for comparison. For the Freescale PA, the DNN model is used for comparison experiment. The comparison figures show that the Wiener‐type DNN model has better convergence properties than the DNN model in the nonlinear region of the PA. The established accurate PA model is conducive to the design of subsequent communication circuits.

Anisotropic model observing pulsars from Neutron Star Interior Composition with modified Van der Waals equation of state

This paper is designed for heavy pulsars coming from the Neutron Star Interior Composition Explorer. The research model is describe by Einstein field equations for anisotropic fluid configuration with spherical symmetry. As per present perceptiveness, modified non-linear Van der Waals equation of state is used to relate physical variables. The continuity of inner and outer matter is obtained by comparing inner spacetime to outer Schwarzschild metric. The physical viability of this model is evaluated and further it is compared with observational data of pulsars PSR J0348+0432, PSR J0740+6620 and PSR J0030+0451. The model fulfils all physical and mathematical characteristics of the dense structure studies. It offers the factual proofs carried by evolution of celestial configurations. The working model presented here is physically viable and shows stable behaviour.

Interval observers for hybrid continuous-time stationary systems

The problem of interval observer design for hybrid continuous-time stationary systems under external disturbances and measurement noise is studied. It is assumed that continuous-time dynamic of such systems is described by linear or nonlinear differential state equations with linear function of output. System parameters depend on system states and are switched based on the control system with finite set of states. The relations allowing designing hybrid interval observer of minimal dimension estimating the set of admissible values of the prescribed linear vector function of the system states are derived. To solve the problem, algebra of partitions and linear algebra are used. Theoretical results are illustrated by example.

A novel unified variational finite difference (VFD) solution method for optimal control problems

This article is aimed to propose a simple yet efficient unified numerical strategy for solving both linear and non-linear optimal control problems. To do so, the general form of quadratic performance index function and nonlinear state equations are considered first. Then, the idea of variational differential quadrature method is used to convert the integral/differential equations to the equivalent algebraic form. Since time is the only independent variable is this research, the finite difference method with an equally-spaced discretization scheme would be a more appropriate technique rather than the differential quadrature approach. So, the implemented numerical solution is called now as the variational finite difference method. The method of Lagrange multipliers is then utilized for minimization purpose and, as a result, the final set of nonlinear algebraic equations are obtained. Finally, the quadratic and triadic forms of non-linearity are considered and an explicit formulation is represented for the residual and Jacobian of the Newton-based iterative solution procedure. To demonstrate the accuracy and efficiency of the proposed approach in the quadratic optimal control area, several benchmark problems involving linear time-invariant, linear time-variant and nonlinear examples are successfully solved and the results are confirmed with those existed in the literature.

Dynamical stability and phase space analysis of an emergent Universe with non-interacting and interacting fluids

Abstract We investigate the evolution of a flat Emergent Universe obtained with a non-linear equation of state (nEoS) in Einstein’s general theory of Relativity. The nEoS is equivalent to three different types of barotropic cosmic fluids, which are found from the nEoS parameter. The EU began expanding initially with no interactions among the cosmic fluids. However, one can assume an interaction that sets in at a time t ⩾ t i in the cosmic fluids, we study the evolution of the EU that accommodates the present observed Universe. We adopt dynamical system analysis to study the critical points of the autonomous system and the evolutionary behavior of an EU with or without interaction among the cosmic fluid components. We also study the stability of critical points and plot the phase portraits. The density parameters and the corresponding cosmological parameters are determined for both the non-interacting and interacting phases of the evolutionary dynamics of the Universe.

Observational constraints on the emergent universe with non-linear equation of state and interacting fluids

We investigate a flat emergent universe (EU) with a nonlinear equation of state equivalent to three different composition of fluids. In the EU initially, the evolution of the universe began with no interaction but as time evolves an interaction sets in among the three fluids leading to the observed universe. The characteristic of an EU is that it is a singularity free universe that evolves with all the basic features of the early evolution. For a given nonlinear equation of state parameter, it permits a universe with three different fluids, we get a universe with dark energy, cosmic string, and radiation domination to begin with which at a later epoch transits into a universe with three different fluids with matter domination (baryonic as well as dark matter) and dark energy for a given interaction strength among the cosmic fluids. The evolution of the universe is probed with exponential interactions to obtain a universe with late acceleration. The model parameters are constrained using the observed Hubble data and Type Ia Supernova (SnIa) data from the Pantheon data set. An interacting EU transforms to a matter dominated phase with DE accommodating the present universe satisfactorily. The stability of the cosmological model is also discussed.

ПРИНЦИП НАЛОЖЕНИЯ ДЕФОРМАЦИЙ В ТЕОРИИ ЖЕЛЕЗОБЕТОНА

The work substantiates the correctness of applying the principle of the overlay of deformations when deriving equations of the mechanical state of concrete that are significant in the theory of reinforced concrete in linear and nonlinear formulations. This principle consists in the summation at some point in time of partial deformations corresponding to partial increments of non-decreasing stress at successive previous points in time. Strains are represented by the sum of instantaneous strains and delayed creep strains at the moment of application. The superposition of partial creep strains is implemented according to the Boltzmann superposition principle known in the linear theory of creep - each partial increment of creep strain depends only on the magnitude and duration of the generating partial stress increment and does not depend on its other increments. When taking aging into account, postulated by this principle the mutual independence of partial increments of creep strain takes place relative to partial increments of the stress level. Representing the deformation as the sum of the initial elastic deformation and the creep deformation with a measure that takes into account the evolution of the elastic modulus leads to a different type of equation of state, which is perceived as incorrect in some publications. This perception of a different type of equation of state together with constructions, not related to the principle of the overlay of deformations, led these authors to declare that the principle of the overlay of deformations in the theory of creep is erroneous. In contrast to the traditional approach, a material (concrete, steel, wood, plastic) is considered as a combination of fractions (fibers, layers) with statistically distributed strengths. The concept of strength structure makes it possible to substantiate the principle of superimposing deformations in a nonlinear formulation. In this case, the structure of the nonlinear equation of state turns out to be identical to the structure of the linear equation. This circumstance is significant in applications, for example, when solving relaxation problems that are important for the long-term safety of building structures.

Modulation instability analysis, and characterize time-dependent variable coefficient solutions in electromagnetic transmission and biological field

This study integrates the telegraph model with the traveling wave coefficient. This model characterizes planar random motion of particles in fluid flow, pressure waves from pulsatile blood flow in arteries, electrical impulses in nerve and muscle cell axons, and electromagnetic waves in superconducting media. Using the efficient and well-established Enhanced Modified Simple Equation (EMSE) method, we investigate solitary wave solution with the variable coefficient of the telegraph model. To investigate the variable coefficient solitary wave solution, we apply a transformation variable η = h(t)x + g(t), which is dependent on the time variable function. The diverse functions of h(t) and g(t) yield many novel and exclusive solutions. Numerical solutions are plotted in 3D, density, and 2D. instanton soliton, kinky periodic wave, lump wave, kink wave, anti-kink wave, double periodic wave, diverse types periodic wave, lump with bell wave periodic wave, periodic lump wave, etc. Solitary waves explain complex wave phenomena through nonlinear effects like self-interaction and dispersion. Due to the classical nonlinear telegraph model's wave dynamics, they are used in brain function and communication system design. We conclude by testing the governing model's modulation instability. Dispersive and nonlinear processes modulating the stable state cause instability in high-order nonlinear equations. Examining the wave movement role and modulation instability analysis to assess solution stability shows that all solutions are accurate and stable. The computational difficulties and results demonstrate the approaches' clarity, effectiveness, and simplicity, suggesting they can be applied to evolutionary dynamic and static nonlinear equations in computational physics, other real-world situations, and numerous academic disciplines.

Reliability and sensitivity analysis of delamination growth of composite laminate structures using two efficient sampling methods

In recent years, composite structures have been used in a large number of applications in aerospace, machinery, marine, and civil engineering. However, there are inevitably many uncertainties in the whole life cycle of composite structures, which can easily lead to structural damage and failure. Therefore, it is important to analyze the reliability and sensitivity of composite structures. At present, most of the contributions use the first-order reliability method (FORM) and the second-order reliability method (SORM) to study the reliability of composite structures and compare them with the results of the Monte Carlo simulation. However, both methods have their limitations. FORM cannot guarantee the calculation accuracy for the highly nonlinear limit state equation, and the calculation efficiency of SORM is too low. Therefore, this paper proposes to use importance sampling (IS) and backpropagation neural network-based Monte Carlo (MC-BPNN) to study the reliability, sensitivity, and dispersion of delamination growth of composite laminates. The results show that compared with FORM and SORM, IS and MC-BPNN have higher calculation accuracy and efficiency and can more accurately evaluate the failure degree of composite structures and ensure their safe operation in the field of aerospace equipment. The universality of this method can make it being widely used in the reliability and sensitivity analysis of different composite materials as well as dispersion analysis.

A multiscale model to predict the equation of state of a saturated cement paste

An advanced multiscale analytical approach for predicting the Equation of State (EOS) of fully saturated cement pastes is presented. This approach models the matrix as an elastic–plastic material with linear hardening that includes capillary pores filled with water. The model takes into account both the stochastic variation of pore sizes and their spatial distribution. The microscale level is represented by a spherical medium having the mechanical properties of the cement paste matrix. A single spherical pore that is filled with compressible water is located at the center of the micro-domain. The behavior of the compressible water is modeled by the nonlinear Tait equation of state. The EOS at the macro level is obtained by averaging the micro level domain strains over the pore sizes and pore wall thicknesses. It is shown that the solution depends only on the ratio between the minimum pore wall thickness and the lower limit of capillary pore radius and is independent of each of these parameters separately. The EOS obtained by the proposed model shows good agreement with experimental data for cement paste with water/cement ratio w/c = 0.50. The performed parametric study shows that the bulk modulus and yield stress of undisturbed cement gel matrix significantly affect the EOS, while the variation of Poison ratio, plastic hardening parameter and minimum limit pore size and wall thickness (within physically accepted range) slightly affect the cement paste bulk behavior.

An algorithm to transform discrete-time state equations into the extended observer form

The problem of transforming a single-input single-output nonlinear discrete-time state equations into the extended observer form is studied in this paper. Unlike previous solutions which compute the injection terms from the input–output equations, in this paper the computations utilize directly the given state equations. An algorithm is given to find step-by-step the necessary transformation by computing backward-shifts of functions computed in previous step. It is shown that the integrability conditions presented in the algorithm are necessary and sufficient for solvability of the problem. Various examples have been studied to demonstrate the usefulness of the algorithm.

Cosmologies with positive λ: hierarchies of future behaviour.

Smooth Cauchy data on [Formula: see text] for the Einstein-[Formula: see text]-vacuum field equations with cosmological constant [Formula: see text] that are sufficiently close to de Sitter data develop into a solution that admits a smooth conformal boundary [Formula: see text] in its future. The conformal Einstein equations determine a smooth conformal extension across [Formula: see text] that defines on 'the other side' again a [Formula: see text]-vacuum solution. In this article, we discuss to what extent these properties generalize to the future asymptotic behaviour of solutions to the Einstein-[Formula: see text] equations with matter. We study Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions and the Einstein-[Formula: see text] equations coupled to conformally covariant matter transport equations, to conformally privileged matter equations, and to conformally non-covariant matter equations. We present recent results on the Einstein-[Formula: see text]-perfect-fluid equations with a nonlinear asymptotic dust or asymptotic radiation equation of state. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Investigation of soliton solutions to the truncated M-fractional (3+1)-dimensional Gross-Pitaevskii equation with periodic potential

<p>This research explores some modernistic soliton solutions to the (3+1)-dimensional periodic potential the Gross–Pitaevskii equation with a truncated M-fractional derivative plays a significant role in Bose–Einstein condensation, which describes the dynamics of the condensate wave function. The obtained results include trigonometric, hyperbolic trigonometric and exponential function solutions. Three techniques named: the $ \exp_a $ function method, the Sardar sub-equation method, and the extended $ (G'/G) $-expansion approach are employed to achieve a variety of new solutions for the governing model. More comprehensive information about the dynamical representation of some of the solutions is being presented by visualizing the 2D, 3D and contour plots. This work reveals a number of new types of traveling-wave solutions, such as the double periodic singular, the periodic singular, the dark singular, the dark kink singular, the periodic solitary singular, and the singular soliton solutions. These novel solutions are not the same as those that were previously studied for this governing equation. The presented techniques demonstrate clarity, efficacy, and simplicity, revealing their relevance to diverse sets of dynamic and static nonlinear equations pertaining to evolutionary events in computational physics, in addition to other real-world applications and a wide range of study fields for addressing a variety of other nonlinear fractional models that hold significance in the fields of applied science and engineering.</p>

Anisotropic Strange Stars with Nonlinear Equation of State

The modeling of superdense matter is an interesting research area and in the last decades, such models allow explain the behavior of massive objects as neutron stars, quasars, pulsars and white dwarfs. Taking local anisotropy into consideration, in this paper we present a new classes of solutions for the Einstein´s field equations in a spherically symmetric spacetime under the influence of an electric field considering a quadratic equation of state with a particular form the metric potential that depends on an adjustable parameter. The obtained solutions can be written in terms of elementary functions, namely polynomials and algebraic functions. The graphs generated show that physical variables such as metric potentials, radial pressure, energy density, charge density, anisotropy, radial speed sound are consistent with realistic stellar models. The results of this research can be useful in the development and description of new models of compact structures. Keywords: quadratic equation of state, Einstein-Maxwell system, metric potential, adjustable parameter, compact structures

Analysis of clustered cable-actuation strategies of V-Expander tensegrity structures

This study presents an analysis method to design cable-actuated clustered V-Expander tensegrity elementary cells based on the nonlinear tensegrity statics. Clustered actuation takes advantage of the presence of cable elements in a tensegrity structure by allowing cables to run over frictionless joints. The study first provides the nonlinear static equations of clustered tensegrity structures. Then, it introduces the topology of V-Expander tensegrity elementary cells, which vary in complexity and aspect ratio. To achieve cable actuation, specific sets of cables are chosen from the complete pool of cables to serve as the active elements of V-Expander cells. During actuation, the length of these active cables decreases. The actuation efficiency of each specific choice of active cable is computed to find the optimal actuation strategy. The proposed approach can be utilized to analyze and design other types of cable-driven tensegrity structures, with applications to morphing structures and soft robots.

Improving power system low-frequency oscillations damping based on multiple-model optimal control strategy using polynomial combination algorithm

In this paper, the damping of low-frequency oscillations (LFOs) in a power system containing Static VAR Compensator (SVC) is investigated based on Multiple-Model Linear Optimal Control (MMLOC) Strategy. The performance of MMLOC is enhanced using Polynomial Combination Algorithm (PCA). A linear optimal controller (LOC) generates the optimal control signal with linearization the nonlinear state equations of the system and solving the Riccati equation. One LOC generates synchronized control signals for the synchronous generator excitation system as well as SVC. The LOC is assigned to generate each model’s control signal. The final control signal is made by interpolating the LOC output signals obtained from each model using PCA. Considering a three-phase symmetrical fault on a near bus to the generator, the proposed control strategy is evaluated. This strategy not only maintains stability but also reduces LFO effectively. Furthermore, steady state error of rotor speed and rotor angle tend to zero favorably. Simulations are done for single- and multi-machine power systems via a MATLAB m-code.

A Hybrid Hubspace-RNN based approach for Modelling of Non-Linear Batch Processes

The manuscript addresses the problem of developing a modelling strategy that can accurately capture the dynamics of a non-linear batch process, demonstrated on a uni-axial rotational molding process. To this end, this work presents a strategy that utilizes the nonlinear modeling capability of a recurrent neural network (RNN) model, while leveraging the rigor of subspace-based approaches. The proposed modeling strategy is as follows: a subspace identification algorithm is first utilized on all the training batches to obtain the state sequence for these batches. Then the state sequences from all the training batches and the corresponding input sequences are given as the output and the input to a neural network, respectively. This step essentially builds a non-linear state space model, albeit using the state trajectory identified by the subspace model. The output equation obtained from the above-mentioned subspace identification step along with the newly obtained non-linear state equation is now ready to be used for predicting the output trajectory of the system. The results from the experiments illustrate the improved prediction performance of a model obtained by the proposed hybrid Subspace-RNN approach in comparison to both a subspace-based approach and an RNN-based approach.

Data-Driven Encoding: A New Numerical Method for Computation of the Koopman Operator

This paper presents a data-driven method for constructing a Koopman linear model based on the Direct Encoding (DE) formula. The prevailing methods, Dynamic Mode Decomposition (DMD) and its extensions are based on least squares estimates that can be shown to be biased towards data that are densely populated. The DE formula consisting of inner products of a nonlinear state transition function with observable functions does not incur this biased estimation problem and thus serves as a desirable alternative to DMD. However, the original DE formula requires knowledge of the nonlinear state equation, which is not available in many practical applications. In this paper, the DE formula is extended to a data-driven method, Data-Driven Encoding (DDE) of Koopman operator, in which the inner products are calculated from data taken from a nonlinear dynamic system. An effective algorithm is presented for the computation of the inner products, and their convergence to true values is proven. Numerical experiments verify the effectiveness of DDE compared to Extended DMD. The experiments demonstrate robustness to data distribution and the convergent properties of DDE, guaranteeing accuracy improvements with additional sample points. Furthermore, DDE is applied to deep learning of the Koopman operator to further improve prediction accuracy.

NONLINEAR REFINEMENT OF THE DEFORMATION MODEL OF ORTHOTROPIC MATERIALS, THE RIGIDITY OF WHICH DEPENDS ON THE TYPE OF STRESS STATE

As a result of numerous experimental and theoretical studies, it has been established that when deforming both traditional and new structural materials, which are polymers, composites and synthetic structures used in construction, mechanical engineering, and power engineering devices, complicated mechanical properties manifest themselves.Most of these materials have an orthotropic structure complicated by the dependence of deformation and strength characteristics on the type of stress state, which can be interpreted as deformation anisotropy. These properties contradict the generally accepted theories of deformation. Therefore, a number of models have been specially developed for such materials over the past 56 years, taking into account the complicated properties of materials. However, all of them have disadvantages and certain contradictions with the fundamental rules for constructing equations of state. The previous works of the authors of the presented studies establish general approaches to the construction of energetically nonlinear deformation models of composite materials with recommendations for calculating their constants based on a wide range of experiments.It turned out that the set of necessary experiments should include experiments on complex stress states, most of which are technically unrealizable. In another work of the authors in 2021, using the tensor space of normalized stresses, the deformation potential was formulated in a quasi-linear form, constructed in the main axes of orthotropy of materials, for determining constants, which is sufficient for the data of the simplest experiments. Along with the obvious advantages of the introduced potential, it has one drawback, which is to replace real nonlinear diagrams with direct rays with minimal error. Despite the unconditional adequacy of the quasi-linear potential, the use of this level of approximations leads to quantitative errors.Therefore, simplified nonlinear equations of state for composite materials are proposed here, the simplest experiments are sufficient to determine the material functions of which. These equations are based on the general laws of mechanics, on the basis of which the constants of material polynomials for a carbon-graphite composite are calculated, taking into account the Drucker constraints.

The key role of Lagrangian multiplier in mimetic gravitational theory in the frame of isotropic compact star

Recently, the mimetic gravitational theory has gained much attention in the frame of cosmology as well as in the domain of astrophysics. In this study, we show that in the frame of mimetic gravitation theory we are not able to derive an isotropic model. Therefore, we turn our attention to mimetic gravitational theory coupled with the Lagrangian multiplier. The field equations of a static isotropic gravitational system that controls the geometry and dynamics of star structure are studied in the frame of mimetic theory coupled with a Lagrangian multiplier using a non-linear equation of state. An energy density is assumed from where all the other unknowns are fixed and a new isotropic model is derived. The physical analysis of this model is studied from different viewpoints and consistent results compatible with a realistic isotropic star are investigated analytically and graphically. Finally, we show that the model under consideration is stable using the adiabatic index procedure.